Optimal. Leaf size=67 \[ \frac {4 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{3 f}-\frac {2 b \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}} \]
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Rubi [A] time = 0.06, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2627, 3771, 2641} \[ \frac {4 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{3 f}-\frac {2 b \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2627
Rule 2641
Rule 3771
Rubi steps
\begin {align*} \int \sqrt {b \sec (e+f x)} \sin ^2(e+f x) \, dx &=-\frac {2 b \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {2}{3} \int \sqrt {b \sec (e+f x)} \, dx\\ &=-\frac {2 b \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {1}{3} \left (2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx\\ &=\frac {4 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{3 f}-\frac {2 b \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 51, normalized size = 0.76 \[ -\frac {\sqrt {b \sec (e+f x)} \left (\sin (2 (e+f x))-4 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right )}{3 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {b \sec \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 123, normalized size = 1.84 \[ -\frac {2 \left (-1+\cos \left (f x +e \right )\right ) \left (2 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )+\cos ^{2}\left (f x +e \right )-\cos \left (f x +e \right )\right ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {\frac {b}{\cos \left (f x +e \right )}}}{3 f \sin \left (f x +e \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (e+f\,x\right )}^2\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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